Stabilizer type X54

Acting group: Aff1(Z12) × Aff1(Z12)
There are exactly 12 groups in this conjugacy class.
The group order is 4.
(T0,S0),
(TI,S4RF),
(T10Q,S6),
(T3IQ,S10RF).
(T0,S0),
(TI,S10RF),
(T10Q,S6),
(T3IQ,S4RF).
(T0,S0),
(T3I,S4RF),
(T6Q,S6),
(T9IQ,S10RF).
(T0,S0),
(T3I,S10RF),
(T6Q,S6),
(T9IQ,S4RF).
(T0,S0),
(T5I,S4RF),
(T2Q,S6),
(T3IQ,S10RF).
(T0,S0),
(T5I,S10RF),
(T2Q,S6),
(T3IQ,S4RF).
(T0,S0),
(T7I,S4RF),
(T10Q,S6),
(T9IQ,S10RF).
(T0,S0),
(T7I,S10RF),
(T10Q,S6),
(T9IQ,S4RF).
(T0,S0),
(T9I,S4RF),
(T6Q,S6),
(T3IQ,S10RF).
(T0,S0),
(T9I,S10RF),
(T6Q,S6),
(T3IQ,S4RF).
(T0,S0),
(T11I,S4RF),
(T2Q,S6),
(T9IQ,S10RF).
(T0,S0),
(T11I,S10RF),
(T2Q,S6),
(T9IQ,S4RF).

Where
S=T=(1,2,3,4,5,6,7,8,9,10,11,12) ,
R=(6,7)(5,8)(4,9)(3,10)(2,11)(1,12) ,
I=(7)(6,8)(5,9)(4,10)(3,11)(2,12)(1) ,
F=Q=(10)(8,12)(7)(5,9)(4)(3,11)(2,6)(1) .

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